Modeling surface topography of state-of-the-art x-ray mirrors as a result of stochastic polishing process: recent developments

Abstract

Recently, an original method for the statistical modeling of surface topography of state-of-the-art mirrors for usage in xray optical systems at light source facilities and for astronomical telescopes [Opt. Eng. 51(4), 046501, 2012; ibid. 53(8), 084102 (2014); and ibid. 55(7), 074106 (2016)] has been developed. In modeling, the mirror surface topography is considered to be a result of a stationary uniform stochastic polishing process and the best fit time-invariant linear filter (TILF) that optimally parameterizes, with limited number of parameters, the polishing process is determined. The TILF model allows the surface slope profile of an optic with a newly desired specification to be reliably forecast before fabrication. With the forecast data, representative numerical evaluations of expected performance of the prospective mirrors in optical systems under development become possible [Opt. Eng., 54(2), 025108 (2015)]. Here, we suggest and demonstrate an analytical approach for accounting the imperfections of the used metrology instruments, which are described by the instrumental point spread function, in the TILF modeling. The efficacy of the approach is demonstrated with numerical simulations for correction of measurements performed with an autocollimator based surface slope profiler. Besides solving this major metrological problem, the results of the present work open an avenue for developing analytical and computational tools for stitching data in the statistical domain, obtained using multiple metrology instruments measuring significantly different bandwidths of spatial wavelengths.

Investigation on lateral resolution of surface slope profilers

“…12 are also relevant to the classical arrangement of a NOM-like slope profiler with an autocollimator ELCOMAT-3000-based sensor with a circular aperture of 2.5-mm diameter [46˗58]. The resolution properties of the OSMS with 2.5-mm aperture are well described with a circular aperture ITF, analytically derived in [68] and corrected in Ref. [66] with an additional term, empirically determined to better match the OSMS resolution measurements with the high-spatial-frequency pattern of the chirped slope test sample [22,23], used also here:…”

“…Some introductory information can be found in Ref. [68]. In this publication, the ITF-based reconstruction (deconvolution) of the measured slope data beyond the instrumental resolution limit was first suggested and demonstrated on the example of a 1D data, forecasted using stochastic modeling of the measured data.…”

Optimization of the size and shape of the scanning aperture in autocollimator-based deflectometric profilometers

“…If the slope profilometer's MTF is known (precisely measured) over the extended frequency range, it can be used for partial slope data restoration over the spatial frequency range above the resolution limit determined by the first zero-crossing of the OTF. 36 The simple model used here predicts the first zerocrossings at the spatial wavelengths of 1.5 mm and 2.05 mm for the optimal tangential and the standard circular apertures, respectively. However, in publications 25 (see also the results of our measurements in Sec.…”

“…Here, we follow to the approach outlined in Ref. 36. The aperture effect is characterized with the point spread function, PSF (see, for example, Refs.…”

“…For an analytical expression for the corresponding 1-D point spread function, see, for example, Ref. 36. The OTF, given with Eq.…”